Effective scalar products of D-finite symmetric functions

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective D-Finite Symmetric Functions”.) Introduction A power series in one variable is called differentiably finite, or simply D-finite, when it is solution of a linear differential equation with polynomial coefficients. This differential equation turns out to be a convenient data structure for extracting information related to the series and many algorithms operate directly on this differential equation. In particular, the class of univariate D-finite power series is closed under sum, product, Hadamard product, and Borel transform, among other operations, and algorithms computing the corresponding differential equations are known (see for instance [1]). Moreover, the coefficient sequence of a univariate D-finite power series satisfies a linear recurrence, which makes it possible to compute many terms of the sequence efficiently. These closure properties are implemented in computer algebra systems [2, 3]. Also, the mere knowledge that a series is D-finite gives information concerning its asymptotic behaviour. Thus, whether it be for algorithmic or theoretical reasons, it is often important to know whether a given series is D-finite or not, and it is useful to compute the corresponding differential equation when possible. D-finiteness extends to power series in several variables: a power series is called D-finite when the vector space spanned by the series and its derivatives is finite-dimensional. Again, this class enjoys many closure properties and algorithms are available for computing the systems of linear differential equations generating the corresponding operator ideals [4, 5]. Algorithmically, the key tool is provided by Gröbner bases in rings of linear differential operators and an implementation is available in Chyzak’s Mgfun package. An additional, very important closure operation on multivariate D-finite power series is definite integration. It can be computed by an algorithm called creative telescoping, due to Zeilberger [6]. Again, this method takes as input (linear) differential operators and outputs differential operators (in fewer variables) satisfied by the definite integral. It turns out that the algorithmic realization of creative telescoping has several common features with the algorithms we introduce here. Beyond the multivariate case, Gessel considered the case of infinitely many variables and laid the foundations of a theory of D-finiteness for symmetric functions [7]. He defines a notion of D-finite symmetric series and obtains several closure properties. The motivation for studying D-finite symmetric series is that new closure properties occur and can be exploited to derive the D-finiteness of usual multivariate or univariate power series. Thus, the main application of [7] is a proof of the D-finiteness for several combinatorial counting functions. This is achieved by describing the counting functions as combinations of coefficients of D-finite symmetric series, which can then be computed by way of a scalar product of symmetric functions. Under certain conditions, the scalar product of symmetric functions depending on extra parameters is D-finite in This package is part of the algolib library available at http://algo.inria.fr/packages/. 1 2 FRÉDÉRIC CHYZAK, MARNI MISHNA, AND BRUNO SALVY those parameters, where D-finiteness is that of (usual) multivariate power series. Most of Gessel’s proofs are not constructive. In this article, we give algorithms that compute the resulting systems of differential equations for the scalar product operation. Besides Gessel’s work, these algorithms are inspired by methods used by Goulden, Jackson, and Reilly in [8, 9]. Finally, Gröbner bases are used to help make these methods into algorithms. One outcome is a simplification of the original techniques of [8, 9]. Considering some enumerative combinatorial problem of a symmetric flavour and parameterized by a discrete parameter (denoted by k in the examples below), it is often so that the enumeration is solved by first forming a scalar product of two symmetric functions in k variables. Moreover, in the examples envisioned (the enumeration of k-regular graphs, of k-uniform tableaux, etc.), this scalar product is the specialization to k variables of a scalar product between two “closed form” symmetric functions in infinitely many variables. Both symmetric functions are sufficiently well-behaved that nice “closed form” are obtained under specialization, leading to descriptions in terms of linear differential operators that are easy to derive. This nice behaviour is well exemplified by Eq. (6) and Eq. (9) below and is what delimits the scope of our method in applications. Additionally, our method extends to more scalar products whose associated adjunctions satisfy a certain condition of preservation of degree (see Section 9.1), as well as to the Kronecker product of symmetric functions (see Section 9.2). This article is organized as follows. After recalling the necessary part of Gessel’s work in Section 1, we start by focusing on the special situation when a single argument of the scalar product depends on extra parameters. We present an algorithm for computing the differential equations satisfied by the scalar product in this case in Section 2. The application to the example of k-regular graphs is detailed in Section 3. Then a special case where the algorithm can be further tuned is described in Section 4. We treat a variant of Young tableaux where each element is repeated k times in Section 5. (These are in bijection with a generalization of involutions [10].) The general form of the main algorithm, when both arguments depend on extra parameters, is given in Section 6. Termination and correctness of the main algorithms are proved in Section 7. Next, in Section 8 we employ our algorithms to derive asymptotic estimates of the enumerating sequences of k-regular graphs for k = 1, 2, 3, 4. Following this approach of experimental mathematics, we state a conjecture for general k. A discussion on several extensions and applications of the method closes the paper in Section 9, including the calculation of a seemingly new formula for the Kronecker product of the sum of the Schur functions with itself. 1. Symmetric D-finite Functions To begin we recall the facts we need about symmetric functions, D-finite functions, and symmetric D-finite functions. 1.1. Symmetric functions. We refer the reader to [11, 1] for further definitions, notation, and results related to symmetric functions. Denote by λ = (λ1, . . . , λk) a partition of the integer n. This means that n = λ1 + · · · + λk and λ1 ≥ · · · ≥ λk > 0, which we also denote λ ` n. Partitions serve as indices for the five principal symmetric function families that we use: homogeneous (hλ), power (pλ), monomial (mλ), elementary (eλ), and Schur (sλ). These are series in the infinite set of variables, x1, x2, . . . over a field K of characteristic 0. When the indices are restricted to all partitions of the same positive integer n, any of the five families forms a basis for the vector space of symmetric polynomials of degree n in x1, x2, . . . On the other hand, the family of pi’s indexed by the integers i ∈ N generates the algebra Λ of symmetric functions over K: Λ = K[p1, p2, . . . ]. Furthermore, the pi’s are algebraically independent over Z. Generating series of symmetric functions live in the larger ring of symmetric series, K[t][[p1, p2, . . . ]]. There, we have the generating series of homogeneous and elementary functions: